A simple point process is a special type of point process in probability theory. In simple point processes, every point is assigned the weight one.
Definition
Let S be a locally compact second countable Hausdorff space and let \mathcal S be its Borel \sigma-algebra. A point process \xi, interpreted as random measure on (S, \mathcal S), is called a simple point process if it can be written as
\xi =\sum_{i \in I} \delta_{X_i}
for an index set I and random elements X_i which are almost everywhere pairwise distinct. Here \delta_x denotes the Dirac measure on the point x.
Examples
Simple point processes include many important classes of point processes such as Poisson processes, Cox processes and binomial processes.
Uniqueness
If \mathcal I is a generating ring of \mathcal S then a simple point process \xi is uniquely determined by its values on the sets U \in \mathcal I. This means that two simple point processes \xi and \zeta have the same distributions iff
P(\xi(U)=0) = P(\zeta(U)=0) \text{ for all } U \in \mathcal I
Literature
- Kallenberg, Olav (2017). Random Measures, Theory and Applications. Vol. 77. Probability Theory and Stochastic Modelling. Switzerland: Springer. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
- Daley, D.J. & Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods. New York: Springer. ISBN 0-387-95541-0.